Among the ways to define Chern classes for vector bundles in algebraic topology, defining them by pulling back cohomology classes from the classifying space $BU(n)$ is one elegant way to do so.
In algebraic geometry there is no classifying scheme, but one can define the stack $M$ of vector bundles over schemes. I don't know anything about this stack.
Question: Is there a reasonable notion of Chow ring for $M$ so that the algebraic chern classes are obtained by pulling back cycles from $M$ along classifying maps? Or is this for some reason the wrong question?
(By algebraic Chern classes I mean what we obtain by formally inverting the generating function built from the Segre classes of a vector bundle, which are an endomorphism of the chow ring obtained by pulling back, intersecting with the canonical divisor in the bundle some number of times, and then pushing forward. I think these algebraic Chern classes can also be defined by twisting a vector bundle until it has enough sections, and then looking at the vanishes scheme of some the minors.)